Physical Sciences and Mathematics Commons^™

Computer Assistance In Discovering Formulas And Theorems In System Engineering, J. W. Helton, Mark Stankus

Mathematics

If one reads a typical article on A,B,C,D systems in the control transactions, one finds that most of the algebra involved is non commutative rather than commutative. Thus, for symbolic computing to have much impact on linear systems research, one needs a program which will do non-commuting operations. Mathematica, Macsyma and Maple do not. We have a package, NCAlgebra, which runs under Mathematica which does the basic operations, block matrix manipulations and other things. The package might be seen as a competitor to a yellow pad. Like Mathematica the emphasis is on interaction with the program and flexibility.

The issue …

M-Isometric Transformations Of Hilbert Space, I, Jim Alger, Mark Stankus

Mathematics

No abstract provided.

A Statistical Derivation Of The Significant-Digit Law, Theodore P. Hill

Research Scholars in Residence

The history, empirical evidence and classical explanations of the significant-digit (or Benford's) law are reviewed, followed by a summary of recent invariant-measure characterizations. Then a new statistical derivation of the law in the form of a CLT-like theorem for significant digits is presented. If distributions are selected at random (in any "unbiased" way) and random samples are then taken from each of these distributions, the significant digits of the combines sample will converge to the logarithmic (Benford) distribution. This helps explain and predict the appearance of the significant0digit phenomenon in many different empirical contexts and helps justify its recent application …

Adjacencies In Words, Jean-Marc Fedou, Don Rawlings

Mathematics

Based on two inversion formulas for enumerating words in the free monoid by adjacencies, we present a new approach to a class of permutation problems having Eulerian-type generating functions. We also show that a specialization of one of the inversion formulas gives Diekert's lifting to the free monoid of an inversion theorem due to Cartier and Foata.

The Significant-Digit Phenomenon, Theodore P. Hill

Research Scholars in Residence

No abstract provided.

M-Isometric Transformations Of Hilbert Space, Ii, Jim Alger, Mark Stankus

Mathematics

No abstract provided.

Base-Invariance Implies Benford's Law, Theodore P. Hill

Research Scholars in Residence

A derivation of Benford's Law or the First-Digit Phenomenon is given assuming only base-invariance of the underlying law. The only base-invariant distributions are shown to be convex combinations of two extremal probabilities, one corresponding to point mass and the other a log-Lebesgue measure. The main tools in the proof are identification of an appropriate mantissa σ-algebra on the positive reals, and results for invariant measures on the circle.